Graph Theory: Lecture No. 1 - NPTel

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1 Reinhard Diestel : Graph Theory (Springer). 2 Douglas B. West: Introduction to Graph Theory. (Prentice-Hall India). 3 A. Bondy and U.S.R. Murty: Graph Theory ...
Graph Theory: Lecture No. 1

Graph Theory: Lecture No. 1 L. Sunil Chandran Computer Science and Automation, Indian Institute of Science, Bangalore Email: [email protected]

Graph Theory: Lecture No. 1

References

1

Reinhard Diestel : Graph Theory (Springer)

2

Douglas B. West: Introduction to Graph Theory (Prentice-Hall India)

3

A. Bondy and U.S.R. Murty: Graph Theory (Springer)

4

B. Bollabas : Modern Graph Theory (Springer)

Graph Theory: Lecture No. 1

What is a Graph ?

It is a triple consisting of a vertex set V (G ), an edge set E (G ) and a relation that associates with each edge two vertices (not necessarily distinct) called its end points.

Graph Theory: Lecture No. 1

A loop Multiple edge. Simple Graph Finite Graph

Graph Theory: Lecture No. 1

Some simple graphs

Complete Graph. Cycle Path

Graph Theory: Lecture No. 1

Subgraph and Induced Subgraph

H is a subgraph of G : Then V (H) ⊆ V (G ), E (H) ⊆ E (G ). The assignment of end points to edges in H is the same as that in G . H is an induced subgraph of G on S, where S ⊆ V (G ): Then V (H) = S and E (H) is the set of edges of G such that both the end points belong to S.

Graph Theory: Lecture No. 1

A graph G is connected if each pair of vertices belongs to a path.

Graph Theory: Lecture No. 1

Isomorphism

An Isomorphism from a simple graph G to a simple graph H is a bijection f : V (G ) → V (H) such that (u, v ) ∈ E (G ) if and only if (f (u), f (v )) ∈ E (H)

Graph Theory: Lecture No. 1

Forest and Tree

A graph without any cycle is acyclic. A forest is an acyclic graph A tree is a connected acyclic graph.

Graph Theory: Lecture No. 1

Bipartite Graph

A graph G is bipartite if V (G ) is the union of two disjoint (possibly empty) independent sets called partite sets of G . (A subset S ⊆ V (G ) is an independent set if the induced subgraph on S contains no edges. )

Graph Theory: Lecture No. 1

A tree is a bipartite graph. Can we say that the complete graph Kn is bipartite ? The complete bipartite graph.

Graph Theory: Lecture No. 1

Vertex Cover

A set S ⊆ V (G ) is a vertex cover of G (of the edges of G ) if every edge of G is incident with a vertex in S. A vertex cover of the minimum cardinality is called a minimum vertex cover. We will denote this set by MVC(G).

Graph Theory: Lecture No. 1

What is the cardinality of MVC in the complete graph Kn ? What about the complete bipartite graph Km,n ? The cycle Cn , when n is even and odd ?

Graph Theory: Lecture No. 1

The cardinality of a biggest independent set in G is called the independence number (or stability number) of G and is denoted by α(G ).

Graph Theory: Lecture No. 1

Is there any relation between |MVC (G )| and α(G ) ? If we remove a VC from G , the rest is an independent set. So, if we remove MVC from G , the rest, i.e. V − MVC is an independent set. So, α(G ) ≥ n − |MVC (G )|. Thus |MVC (G )| ≥ n − α(G ). Similiarly if we remove any independent set from G , the rest is VC, and so |MVC | ≤ n − α(G ). Thus we get |MVC | = n − α(G ). If we denote —MVC(G)— by β(G ), then we have β(G ) + α(G ) = n.

Graph Theory: Lecture No. 1

Matching

A set M of independent edges in a graph is called a matching. M is a matching of U ⊆ V (G ), if every vertex in U is incident with an edge in M. Then a vertex in U is a matched vertex. The vertices which are not incident with any edge of M is unmatched.

Graph Theory: Lecture No. 1

A matching M is a perfect matching of G , if every vertex in G is matched by M. If G has n vertices, what is the cardinality of a perfect matching M of G ?

Graph Theory: Lecture No. 1

The cardinality of the biggest matching in G can be denoted by α0 (G ).

Graph Theory: Lecture No. 1

What is the value of α0 (G ) for: Cycle Cn Path Pn Complete Graph Kn Complete Bipartite graph Km,n